Dynamic correlations and possible diffusion pathway in the superionic conductor Cu2−x Se

Very high-resolution X-ray scattering data are used to study the disordered structure of Cu2−x Se in the high-temperature superionic phase.


Effects of Punch type and size
The method used to separate Bragg peaks from the diffuse scattering has an effect on the resulting 3D-ΔPDF.
If too little intensity is removed from the Bragg positions, positive peaks will be seen for all interatomic vectors in the 3D-ΔPDF, as this is essentially the same as partially adding the Patterson function to the 3D-ΔPDF. This will also result in a large positive integral over the whole 3D-ΔPDF. If too much scattering is removed around the Bragg positions, it will typically result in the addition of spurious negative peaks to the 3D-ΔPDF, which can be hard to distinguish from negative features in the 3D-ΔPDF, which are real. In cases where the diffuse scattering peaks at the Bragg position, removal of too much diffuse scattering will also result in a too fast decay of the 3D-ΔPDF, as the high-frequency components of a diffuse peak are removed. Figure S3 shows the total scattering in the HK0 and HHL planes, as well as the scattering obtained after punch and fill and the resulting 3D-ΔPDFs. The top row shows the total scattering, with strong and sharp Bragg peaks at symmetry-allowed integer H, K and L positions, as well as spread-out diffuse scattering. The diffuse scattering peaks at the Bragg positions, and is particularly strong around the [2,2,0] , [4,0,0] and [4,4,0] reflections. The second row shows the result of using a small punch on all allowed Bragg positions of the space-group. This removes most of the Bragg scattering, but leaves very strong peaks at positions such as [1,1,1], suggesting that the small punch size was not sufficient to catch all Bragg peaks. The third row shows the result of using a larger punch size. Although this removes all Bragg peaks, it is also clear that it removes too much of the diffuse scattering. E.g. there is a clear problem with the [2,2,0] peaks, where too much has been removed at the Bragg position, leaving a small "pinch" after filling. The 3D-ΔPDF now shows negative peaks at positions which were not negative before. As too much scattering around Bragg peaks was removed, these negative peaks are probably not real, but a consequence of having "holes" in the scattering at the Bragg positions. The fourth row shows the result of using a frame-by-frame removal of sharp and intense features as described in the experimental section. This removes much of the Bragg scattering, but is not effective at removing weak peaks due to the necessity of a threshold value to not remove noise. The 3D-ΔPDF in this case now has positive peaks at many interatomic vectors, further showing that some Bragg intensity is still left. The bottom row shows the combination of using the frame-by-frame peak removal together with the small punch size, to catch the remaining weak peaks left after the frame-by-frame peak removal. This produces scattering data where the Bragg peaks have been removed, but without clear "pinches" or holes at the Bragg positions. It One should be cautious when interpreting parts of the 3D-ΔPDF which depend of the punch size and method.
In this case the features interpreted in the paper are those which do not change with the punch size and method.

Problem with overlap of 3D-ΔPDF peaks in determining integral amplitudes
Due to a large overlap of features in the 3D-ΔPDF, it was not possible to isolate the substitutional contribution by integration of features. The top row of Figure S4 shows an attempt at making integration basins around features. Here, all points nearest to an interatomic vector of the ideal antiflourite structure is assigned to the integration basin of that feature. Black lines are used to illustrate the boundary of features.
A clear illustration of the problem of overlap in seen for the features at (0,0,½) and (1/4,1/4,1/4), visible in the HHL plane (top right in Figure S4). Both of these features have negative contributions towards the center of space. The results of integration of these basins is shown in the bottom part of the figure. The feature at (0,0,½) is positive and the feature at (1/4,1/4,1/4) is negative. The (1/4,1/4,1/4) vector is the vector between Cu and Se. If real, the negative integral of the (1/4,1/4,1/4) feature would suggest that this vector separates fewer atoms than in the average structure. This could only happen if there was missing Se in the structure, and Cu and Se vacancies tended to avoid each other. However, there is no indication that there should be any missing Se, and the negative integral of this peak is more likely a result overlap of features, such that the integration basin of (1/4,1/4,1/4) has absorbed much of the negative lobe of the (0,0,½) feature.

Maximum Entropy Method
Fourier inversion shows very weak channels in the structure between Cu sites. To get a better estimate on the uncertainty on these channels, Maximum Entropy Method (MEM) is used. MEM gives the electron density of the unit cell based on the phased structure factors and their errors. It is not dependent on a structural model other than through the phasing of reflections, which is this case is robust for all models. MEM gives the most unbiased electron density consistent with the structure factors (Sakata & Sato, 1990). MEM calculations were performed in the BayMEM software (Smaalen et al., 2003) with the Sakato-Sato algorithm (Sakato & Sato, 1990). A grid of 128×128×128 voxels was used together with a flat prior density. Convergence was obtained at 1. Figure S5 shows the electron density in the xxz plane obtained from Fourier Inversion (FI) and MEM. While the FI density shows very weak channels between Cu sites, the MEM density does not. FI will usually be biased if not all non-zero reflections are included. But in this case the scattering measurement extended much further than any observable reflections as discussed in the main text. This suggests that the channels seen in the FI has a large uncertainty, as the 1 MEM does not contain them. MEM converged to a lower stopping value of will eventually produce the same features as the FI density. Using 1 as a stopping value assumes that the estimated standard deviations (ESDs) are correct. In reality this is not always the case, and if ESDs are overestimated, a 1 MEM density will produce a more smeared-out density compared to the real density.
The MEM density strongly indicates that there is a large uncertainty on the channels observed in the FI density, further corroborating that jumps between Cu sites are rare and that Cu does not move as liquid-like ions through the structure.